发表机构
Sardar Vallabhbhai National Institute of Technology, Surat（萨达尔·瓦拉巴伊国家技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究 d 正则图签名，通过对使短偶圈不平衡的仿射\(\mathbb{F}_2\)签名族求平均，将比卢 - 利尼亚尔猜想符号问题转化为计数问题，证明相关上下界及结论，还确定假设、给出超立方体证书并记录障碍。
AI 中文摘要
对于 d 正则图的签名σ，\(A_σ\)的谱仅取决于圈的符号。我们研究使每个短偶圈不平衡的仿射\(\mathbb{F}_2\)签名族，并表明对其求平均将比卢 - 利尼亚尔猜想的符号问题转化为计数问题：一个主恒等式将族平均迹表示为限制在约束圈跨度\(W\)内的环绕类上的奇偶加权和，并且族平均伊哈拉\(L\)函数对角化，使得每个奇偶性超出\(W\)的素数自动贡献拉马努金速率\(\sqrt{d - 1}\)。相比之下，对所有签名进行均匀平均不能证明谱半径低于凯斯滕轮廓。我们证明了限制游走计数的匹配上下界，一个从下方的加倍注入，以及从上方的耳分解编码，其中非回溯游走的新运行次数等于其支撑的圈秩，结合无自行车图的窗口引理和通过不规则图的摩尔界的秩界。结果包括比卢 - 利尼亚尔猜想的\(\varepsilon\)版本：每个在尺度\(\log n\)下亚临界的 d 正则图，以及每个在半径\(C\log\log n / \delta\)下无自行车的 d 正则图，在奇偶性族中都有一个签名，使得\(\rho(A_σ)\leq2\sqrt{d - 1}(1 + C\delta\log(1 / \delta))(1 + o(1))\)。我们进一步精确确定了必要假设（\(K_d\)捕获；树突发小工具），给出了超立方体上的精确证书，并记录了双边交错的决定性障碍：\(\mathbb{E}_σ\det(xI - A_σ^2)\)不是实根，对于四边形已经如此，此时它等于\((x^2 - 4x + 2)^2 + 4\)。
英文摘要
We develop an affine $\mathbb F_2$ framework for structured signings of regular graphs. A family-averaging identity converts even spectral moments into parity-weighted closed-walk counts supported on the span of prescribed short even cycles, while a kernel-averaged Ihara identity gives the corresponding decomposition at the non-backtracking level. We give a finite-scale bounded-rank counting estimate and a conditioning corollary showing that, on bicycle-free graph sequences, any parity family of uniformly bounded codimension contains near-Ramanujan signings whenever the corresponding random-signing theorem applies. The latter is a transfer statement rather than a new concentration theorem. Finally, on $C_n(1,2)$ for even $n\ge10$, the quadrilateral-unbalanced family has exactly four switching classes and its twisted classes attain $ρ_-(n)=2\sqrt{\cos^2(π/n)+\cos^2(2π/n)}$; a period-$8$ signing has spectral radius $r_*=2.793604493334841\ldots$ for every positive multiple of $8$. Thus for $n=8m\ge32$ the constrained minimum is strictly larger than a value attained by an unrestricted signing, while equality of $r_*$ with the unrestricted minimum remains conjectural.
Comments20 pages
